↑

Properties

Label 2-2e2-4.3-c14-0-5
Degree $2$
Conductor $4$
Sign $-0.595 - 0.803i$
Analytic cond. $4.97315$
Root an. cond. $2.23005$
Motivic weight $14$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−114. − 57.5i)2-s − 2.94e3i·3-s + (9.75e3 + 1.31e4i)4-s − 8.16e4·5-s + (−1.69e5 + 3.37e5i)6-s + 1.09e6i·7-s + (−3.56e5 − 2.06e6i)8-s − 3.90e6·9-s + (9.33e6 + 4.70e6i)10-s − 4.90e6i·11-s + (3.88e7 − 2.87e7i)12-s − 7.29e7·13-s + (6.33e7 − 1.25e8i)14-s + 2.40e8i·15-s + (−7.83e7 + 2.56e8i)16-s − 4.69e8·17-s + ⋯
L(s)  = 1  + (−0.893 − 0.449i)2-s − 1.34i·3-s + (0.595 + 0.803i)4-s − 1.04·5-s + (−0.606 + 1.20i)6-s + 1.33i·7-s + (−0.169 − 0.985i)8-s − 0.817·9-s + (0.933 + 0.470i)10-s − 0.251i·11-s + (1.08 − 0.802i)12-s − 1.16·13-s + (0.600 − 1.19i)14-s + 1.40i·15-s + (−0.291 + 0.956i)16-s − 1.14·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.595 - 0.803i)\, \overline{\Lambda}(15-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s+7) \, L(s)\cr =\mathstrut & (-0.595 - 0.803i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4\)    =    \(2^{2}\)
Sign: $-0.595 - 0.803i$
Analytic conductor: \(4.97315\)
Root analytic conductor: \(2.23005\)
Motivic weight: \(14\)
Rational: no
Arithmetic: yes
Character: $\chi_{4} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4,\ (\ :7),\ -0.595 - 0.803i)\)

Particular Values

\(L(\frac{15}{2})\) \(\approx\) \(0.0482606 + 0.0957880i\)
\(L(\frac12)\) \(\approx\) \(0.0482606 + 0.0957880i\)
\(L(8)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (114. + 57.5i)T \)
good3 \( 1 + 2.94e3iT - 4.78e6T^{2} \)
5 \( 1 + 8.16e4T + 6.10e9T^{2} \)
7 \( 1 - 1.09e6iT - 6.78e11T^{2} \)
11 \( 1 + 4.90e6iT - 3.79e14T^{2} \)
13 \( 1 + 7.29e7T + 3.93e15T^{2} \)
17 \( 1 + 4.69e8T + 1.68e17T^{2} \)
19 \( 1 + 6.82e8iT - 7.99e17T^{2} \)
23 \( 1 + 9.17e8iT - 1.15e19T^{2} \)
29 \( 1 + 1.00e10T + 2.97e20T^{2} \)
31 \( 1 - 3.41e10iT - 7.56e20T^{2} \)
37 \( 1 + 4.99e10T + 9.01e21T^{2} \)
41 \( 1 - 1.91e11T + 3.79e22T^{2} \)
43 \( 1 + 3.10e11iT - 7.38e22T^{2} \)
47 \( 1 + 4.62e11iT - 2.56e23T^{2} \)
53 \( 1 + 5.28e11T + 1.37e24T^{2} \)
59 \( 1 + 8.92e11iT - 6.19e24T^{2} \)
61 \( 1 - 3.16e12T + 9.87e24T^{2} \)
67 \( 1 - 5.43e12iT - 3.67e25T^{2} \)
71 \( 1 + 2.22e12iT - 8.27e25T^{2} \)
73 \( 1 + 8.53e12T + 1.22e26T^{2} \)
79 \( 1 - 1.09e13iT - 3.68e26T^{2} \)
83 \( 1 + 8.13e12iT - 7.36e26T^{2} \)
89 \( 1 + 6.18e13T + 1.95e27T^{2} \)
97 \( 1 - 7.86e13T + 6.52e27T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−19.72712330621370710602865575543, −18.95111030704462356330038877629, −17.73388405083904202413838633092, −15.57561546099833064905993557669, −12.60864386956702029770598892223, −11.62882544482688366384804455857, −8.644587059397343090449448580974, −7.11021714590112684153956536734, −2.33576846319239396960642970235, −0.084808949014009681074580034350, 4.31238432905363213096140662244, 7.54544018841696253713390758708, 9.732253766586565544618437961463, 11.07402408870552053170638614448, 14.79555137642194417066022280212, 16.00717165864436257608213759658, 17.17019033835757002093788304834, 19.56138219248722591576015618008, 20.54766051782421825433254532443, 22.72113846526186064354913135301

Graph of the $Z$-function along the critical line